Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations
Jiabao Su Rushun Tian
Communications on Pure & Applied Analysis 2010, 9(4): 885-904 doi: 10.3934/cpaa.2010.9.885
We study weighted Sobolev embeddings in radially symmetric function spaces and then investigate the existence of nontrivial radial solutions of inhomogeneous quasilinear elliptic equation with singular potentials and super-$(p, q)$-linear nonlinearity. The model equation is of the form

$ -\Delta_p u+V(|x|)|u|^{q-2}u=Q(|x|)|u|^{s-2}u, x\in R^N,$

$ u(x) \rightarrow 0,$ as $ |x|\rightarrow\infty. $

keywords: Inhomogeneous quasilinear elliptic equation Sobolev type embedding.
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system
Rushun Tian Zhi-Qiang Wang
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 335-344 doi: 10.3934/dcds.2013.33.335
Consider the following nonlinear elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u - u=\mu_1u^3+\beta uv^2,\ & \hbox{in}\ \Omega\\ -\Delta v - v= \mu_2v^3+\beta vu^2,\ & \hbox{in}\ \Omega\\ u,v>0\ \hbox{in}\ \Omega, \ u=v=0,\ & \hbox{on}\ \partial\Omega, \end{array} \right. \end{equation*}where $\mu_1,\mu_2>0$ are constants and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\leq3$. We study the existence and non-existence of positive solutions and give bifurcation results in terms of the coupling constant $\beta$.
keywords: indefinite system. Positive solutions bifurcations

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